Determine a magnetic Schrödinger operator with a bounded magnetic potential from partial data in a slab (Q2019992)

The paper studies inverse boundary value problems with partial data for the following magnetic Schrödinger operator \[ \mathcal{L}_{A,q}(x, D)=\sum_{j=1}^n(D_j+A_j(x))^2+q(x), \] where \(D_j=i^{-1}\frac{\partial}{\partial_{x_j}}\), and a complex-valued magnetic field \(A=(A_j)_{1\le j \le n}\) and an electric potential \(q\) are compactly supported functions in the infinite slab. It is shown that the magnetic field and the electric potential can be uniquely determined when the Dirichlet and Neumann data are given on either different boundary hyperplanes or on the same boundary hyperplanes of the slab. The proof is based on the complex geometric optics solutions constructed in [\textit{K. Krupchyk} et al., Commun Math Phys. 327, 993--1009 (2014; Zbl 1295.35366)], which are special solutions to the magnetic Schrödinger equation with \(L^\infty\) magnetic and electric potentials in a bounded domain.